Minimizing statistical bias to identify size effect from beam shear database

Zdenek P Bazant, Qiang Yu

Research output: Contribution to journalArticlepeer-review

20 Scopus citations


The existing database for size effect on shear strength of reinforced concrete beams without stirrups has a bias of two types: 1) Most data points are crowded in the small size range; and 2) the means of the subsidiary influencing parameters, such as the steel ratio and shear-span ratio are very different within different intervals of beam size (or beam depth). To minimize Type 2 bias, the database must be properly filtered. To this end, the size range is first subdivided into intervals of constant size ratio. Then, in each size interval, a computer program progressively restricts the range of influencing parameters both from above and from below, until the mean of the influencing parameter values remaining in that interval attains about the same value in all the size intervals. The centroids of the filtered shear strength data within the individual size interval are found to exhibit a rather systematic trend. Giving equal weight to each interval centroid overcomes the Type 1 bias. The centroids can be closely matched by bivariate least-square regression using Balanťs (energetic) size effect law which was proposed for beam shear in 1984 and in detailed form in 2005. This purely statistical inference of minimized bias also supports the previous fracture-mechanics-based conclusion that, for large sizes, the bi-logarithmic size effect plot must terminate with the asymptotic slope of -1/2. Similar filtering of the database gives further evidence for the previous empirical observation that the shear strength of beams is approximately proportional to the 3/8-power of the longitudinal reinforcement ratio.

Original languageEnglish (US)
Pages (from-to)685-691
Number of pages7
JournalACI Structural Journal
Issue number6
StatePublished - Nov 1 2008


  • Failure probability
  • Fracture mechanics
  • Scaling of failure
  • Shear strength
  • Size effect
  • Statistical analysis

ASJC Scopus subject areas

  • Building and Construction
  • Civil and Structural Engineering

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